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</div>		<h1 class="firstHeading">Moment of inertia</h1>
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			<h3 id="siteSub">From Wikipedia, the free encyclopedia</h3>
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			<dl>
<dd><i>This article is about the moment of inertia of a <b>rotating object</b>. For the moment of inertia dealing with bending of a plane, see <a href="http://en.wikipedia.org/wiki/Second_moment_of_area" title="Second moment of area">second moment of area</a>.</i></dd>
</dl>
<p><b>Moment of inertia</b>, also called <b>mass moment of inertia</b> or the <b>angular mass</b>, (<a href="http://en.wikipedia.org/wiki/SI" class="mw-redirect" title="SI">SI</a> units kg m<sup>2</sup>, <a href="http://en.wikipedia.org/wiki/Imperial_Unit" class="mw-redirect" title="Imperial Unit">Former British units</a> slug ft<sup>2</sup>), is the rotational analog of mass. That is, it is the <a href="http://en.wikipedia.org/wiki/Inertia" title="Inertia">inertia</a> of a rigid rotating body with respect to its rotation. The moment of inertia plays much the same role in <a href="http://en.wikipedia.org/wiki/Rotational_motion" class="mw-redirect" title="Rotational motion">rotational dynamics</a> as mass does in basic dynamics, determining the relationship between <a href="http://en.wikipedia.org/wiki/Angular_momentum" title="Angular momentum">angular momentum</a> and <a href="http://en.wikipedia.org/wiki/Angular_velocity" title="Angular velocity">angular velocity</a>, <a href="http://en.wikipedia.org/wiki/Torque" title="Torque">torque</a> and angular <a href="http://en.wikipedia.org/wiki/Acceleration" title="Acceleration">acceleration</a>, and several other quantities. While a simple <a href="http://en.wikipedia.org/wiki/Scalar_%28physics%29" title="Scalar (physics)">scalar</a> treatment of the moment of inertia suffices for many situations, a more advanced <a href="http://en.wikipedia.org/wiki/Tensor" title="Tensor">tensor</a> treatment allows the analysis of such complicated systems as spinning tops and <a href="http://en.wikipedia.org/wiki/Gyroscope" title="Gyroscope">gyroscope</a> motion.</p>
<p>The symbols <span class="texhtml"><i>I</i></span> and sometimes <span class="texhtml"><i>J</i></span> are usually used to refer to the moment of inertia.</p>
<p>Moment of inertia was introduced by <a href="http://en.wikipedia.org/wiki/Euler" class="mw-redirect" title="Euler">Euler</a> in his book <i>a Theoria motus corporum solidorum seu rigidorum</i>
in 1730. In this book, he discussed at length moment of inertia and
many concepts, such as principal axis of inertia, related to the moment
of inertia.</p>
<table id="toc" class="toc" summary="Contents">
<tbody><tr>
<td>
<div id="toctitle">
<h2>Contents</h2>
 <span class="toctoggle">[<a href="javascript:toggleToc()" class="internal" id="togglelink">hide</a>]</span></div>
<ul>
<li class="toclevel-1"><a href="#Overview"><span class="tocnumber">1</span> <span class="toctext">Overview</span></a></li>
<li class="toclevel-1"><a href="#Scalar_moment_of_inertia"><span class="tocnumber">2</span> <span class="toctext">Scalar moment of inertia</span></a>
<ul>
<li class="toclevel-2"><a href="#Definition"><span class="tocnumber">2.1</span> <span class="toctext">Definition</span></a>
<ul>
<li class="toclevel-3"><a href="#Detailed_Analysis"><span class="tocnumber">2.1.1</span> <span class="toctext">Detailed Analysis</span></a></li>
</ul>
</li>
<li class="toclevel-2"><a href="#Parallel_axis_theorem"><span class="tocnumber">2.2</span> <span class="toctext">Parallel axis theorem</span></a></li>
<li class="toclevel-2"><a href="#Composite_bodies"><span class="tocnumber">2.3</span> <span class="toctext">Composite bodies</span></a></li>
<li class="toclevel-2"><a href="#Equations_involving_the_moment_of_inertia"><span class="tocnumber">2.4</span> <span class="toctext">Equations involving the moment of inertia</span></a></li>
</ul>
</li>
<li class="toclevel-1"><a href="#Moment_of_inertia_tensor"><span class="tocnumber">3</span> <span class="toctext">Moment of inertia tensor</span></a>
<ul>
<li class="toclevel-2"><a href="#Definition_2"><span class="tocnumber">3.1</span> <span class="toctext">Definition</span></a></li>
<li class="toclevel-2"><a href="#Derivation_of_the_tensor_components"><span class="tocnumber">3.2</span> <span class="toctext">Derivation of the tensor components</span></a></li>
<li class="toclevel-2"><a href="#Reduction_to_scalar"><span class="tocnumber">3.3</span> <span class="toctext">Reduction to scalar</span></a></li>
<li class="toclevel-2"><a href="#Principal_moments_of_inertia"><span class="tocnumber">3.4</span> <span class="toctext">Principal moments of inertia</span></a></li>
<li class="toclevel-2"><a href="#Parallel_axis_theorem_2"><span class="tocnumber">3.5</span> <span class="toctext">Parallel axis theorem</span></a></li>
<li class="toclevel-2"><a href="#Other_mechanical_quantities"><span class="tocnumber">3.6</span> <span class="toctext">Other mechanical quantities</span></a></li>
</ul>
</li>
<li class="toclevel-1"><a href="#See_also"><span class="tocnumber">4</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1"><a href="#References"><span class="tocnumber">5</span> <span class="toctext">References</span></a></li>
<li class="toclevel-1"><a href="#External_links"><span class="tocnumber">6</span> <span class="toctext">External links</span></a></li>
</ul>
</td>
</tr>
</tbody></table>
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<p><a name="Overview" id="Overview"></a></p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Moment_of_inertia&amp;action=edit&amp;section=1" title="Edit section: Overview">edit</a>]</span> <span class="mw-headline">Overview</span></h2>
<p>The moment of inertia of an object about a given axis describes how
difficult it is to change its angular motion about that axis. For
example, consider two discs (A and B) of the same mass. Disc A has a
larger radius than disc B. Assuming that there is uniform thickness and
mass distribution, it requires more effort to accelerate disc A (change
its angular velocity) because its mass is distributed further from its
axis of rotation: mass that is further out from that axis must, for a
given angular velocity, move more quickly than mass closer in. In this
case, disc A has a larger moment of inertia than disc B.</p>
<div class="thumb tright">
<div class="thumbinner" style="width: 182px;"><a href="http://en.wikipedia.org/wiki/Image:Synchro.jpg" class="image" title="Divers minimizing their moments of inertia in order to increase their rates of rotation."><img alt="Divers minimizing their moments of inertia in order to increase their rates of rotation." src="Moment_of_inertia_files/180px-Synchro.jpg" class="thumbimage" border="0" height="135" width="180"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/Image:Synchro.jpg" class="internal" title="Enlarge"><img src="Moment_of_inertia_files/magnify-clip.png" alt="" height="11" width="15"></a></div>
Divers minimizing their moments of inertia in order to increase their rates of rotation.</div>
</div>
</div>
<p>The moment of inertia of an object can change if its shape changes.
A figure skater who begins a spin with arms outstretched provides a
striking example. By pulling in her arms, she reduces her moment of
inertia, causing her to spin faster (by the conservation of <a href="http://en.wikipedia.org/wiki/Angular_momentum" title="Angular momentum">angular momentum</a>).</p>
<p>The moment of inertia has two forms, a <a href="http://en.wikipedia.org/wiki/Scalar_%28physics%29" title="Scalar (physics)">scalar</a> form <span class="texhtml"><i>I</i></span> (used when the axis of rotation is known) and a more general <a href="http://en.wikipedia.org/wiki/Tensor" title="Tensor">tensor</a> form that does not require knowing the axis of rotation. The scalar moment of inertia <span class="texhtml"><i>I</i></span> (often called simply the "moment of inertia") allows a succinct analysis of many simple problems in <a href="http://en.wikipedia.org/w/index.php?title=Rotational_dynamics&amp;action=edit&amp;redlink=1" class="new" title="Rotational dynamics (page does not exist)">rotational dynamics</a>,
such as objects rolling down inclines and the behavior of pulleys. For
instance, while a block of any shape will slide frictionlessly down a
decline at the same rate, rolling objects may descend at different
rates, depending on their moments of inertia. A hoop will descend more
slowly than a solid disk of equal diameter because more of its mass is
located far from the axis of rotation, and thus needs to move faster if
the hoop rolls at the same angular velocity. However, for (more
complicated) problems in which the axis of rotation can change, the
scalar treatment is inadequate, and the tensor treatment must be used
(although shortcuts are possible in special situations). Examples
requiring such a treatment include gyroscopes, tops, and even
satellites, all objects whose alignment can change.</p>
<p>The moment of inertia can also be called the <b>mass moment of inertia</b> (especially by mechanical engineers) to avoid confusion with the <a href="http://en.wikipedia.org/wiki/Second_moment_of_area" title="Second moment of area">second moment of area</a>, which is sometimes called the moment of inertia (especially by structural engineers) and denoted by the same symbol <span class="texhtml"><i>I</i></span>. The easiest way to differentiate these quantities is through their <a href="http://en.wikipedia.org/wiki/Units_of_measurement" title="Units of measurement">units</a>. In addition, the moment of inertia should not be confused with the <b><a href="http://en.wikipedia.org/wiki/Polar_moment_of_inertia" title="Polar moment of inertia">polar moment of inertia</a></b>, which is a measure of an object's ability to resist <a href="http://en.wikipedia.org/wiki/Torsion_%28mechanics%29" title="Torsion (mechanics)">torsion</a> (twisting).</p>
<p><a name="Scalar_moment_of_inertia" id="Scalar_moment_of_inertia"></a></p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Moment_of_inertia&amp;action=edit&amp;section=2" title="Edit section: Scalar moment of inertia">edit</a>]</span> <span class="mw-headline">Scalar moment of inertia</span></h2>
<p><a name="Definition" id="Definition"></a></p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Moment_of_inertia&amp;action=edit&amp;section=3" title="Edit section: Definition">edit</a>]</span> <span class="mw-headline">Definition</span></h3>
<p>A simple definition of the moment of inertia of any object, be it a point mass or a 3D-structure, is given by:</p>
<dl>
<dd><img class="tex" alt="I = \int r^2 \,dm" src="Moment_of_inertia_files/403ab94a798c5c1c64af3e569e17b603.png"></dd>
</dl>
<p>where</p>
<dl>
<dd><i>m</i> is the mass,</dd>
<dd>and <i>r</i> is the (perpendicular) distance of the point mass to the axis of rotation.</dd>
</dl>
<p><br></p>
<p><a name="Detailed_Analysis" id="Detailed_Analysis"></a></p>
<h4><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Moment_of_inertia&amp;action=edit&amp;section=4" title="Edit section: Detailed Analysis">edit</a>]</span> <span class="mw-headline">Detailed Analysis</span></h4>
<p>The (scalar) moment of inertia of a <a href="http://en.wikipedia.org/wiki/Point_mass" title="Point mass">point mass</a> rotating about a known axis is defined by</p>
<dl>
<dd><img class="tex" alt="I \triangleq  m r^2\,\!" src="Moment_of_inertia_files/d8ce4f4a6548d3623ffe13a4eaf58d85.png"></dd>
</dl>
<p>The moment of inertia is additive. Thus, for a <a href="http://en.wikipedia.org/wiki/Rigid_body" title="Rigid body">rigid body</a> consisting of <span class="texhtml"><i>N</i></span> point masses <span class="texhtml"><i>m</i><sub><i>i</i></sub></span> with distances <span class="texhtml"><i>r</i><sub><i>i</i></sub></span> to the rotation axis, the total moment of inertia equals the sum of the point-mass moments of inertia:</p>
<dl>
<dd><img class="tex" alt="I \triangleq  \sum_{i=1}^{N} {m_{i} r_{i}^2}\,\!" src="Moment_of_inertia_files/189ba79060e13516ff2b17557c23c7f1.png"></dd>
</dl>
<p>For a solid body described by a continuous mass density function ρ(<b>r</b>), the moment of inertia about a known axis can be calculated by <a href="http://en.wikipedia.org/wiki/Integral" title="Integral">integrating</a> the square of the distance (weighted by the mass density) from a point in the body to the rotation axis:</p>
<dl>
<dd><img class="tex" alt="I \triangleq   \iiint_V r^2 \,\rho(\boldsymbol{r})\,dV \!" src="Moment_of_inertia_files/32a728d99b55121c73a20fd8b534fbff.png"></dd>
</dl>
<p>where</p>
<dl>
<dd>V is the volume occupied by the object.</dd>
<dd>ρ is the spatial <a href="http://en.wikipedia.org/wiki/Density" title="Density">density</a> function of the object, and</dd>
<dd><img class="tex" alt="\boldsymbol{r} \equiv (r,\theta,\phi),(x,y,z), or (r,\theta,z)" src="Moment_of_inertia_files/dbe52d88f4dae1603a90262462622737.png">are coordinates of a point inside the body.</dd>
</dl>
<div class="thumb tright">
<div class="thumbinner" style="width: 166px;"><a href="http://en.wikipedia.org/wiki/Image:Moment_of_inertia_disc.png" class="image" title="Diagram for the calculation of a disk's moment of inertia.  Here k is 1/2 and r is the radius used in determining the moment."><img alt="Diagram for the calculation of a disk's moment of inertia.  Here k is 1/2 and r is the radius used in determining the moment." src="Moment_of_inertia_files/Moment_of_inertia_disc.png" class="thumbimage" border="0" height="84" width="164"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/Image:Moment_of_inertia_disc.png" class="internal" title="Enlarge"><img src="Moment_of_inertia_files/magnify-clip.png" alt="" height="11" width="15"></a></div>
Diagram for the calculation of a disk's moment of inertia. Here <i>k</i> is 1/2 and <i>r</i> is the radius used in determining the moment.</div>
</div>
</div>
<p>Based on <a href="http://en.wikipedia.org/wiki/Dimensional_analysis" title="Dimensional analysis">dimensional analysis</a> alone, the moment of inertia of a non-point object must take the form:</p>
<dl>
<dd><img class="tex" alt=" I = k\cdot M\cdot {R}^2 \,\!" src="Moment_of_inertia_files/88b48fb21580149571769738a5817927.png"></dd>
</dl>
<p>where</p>
<dl>
<dd><i>M</i> is the mass</dd>
<dd><i>R</i> is the radius of the object from the center of mass (in some cases, the length of the object is used instead.)</dd>
<dd><i>k</i> is a dimensionless constant called the <i>inertia constant</i> that varies with the object in consideration.</dd>
</dl>
<p>Inertial constants are used to account for the differences in the
placement of the mass from the center of rotation. Examples include:</p>
<ul>
<li><i>k</i> = 1, thin ring or thin-walled cylinder around its center,</li>
<li><i>k</i> = 2/5, solid sphere around its center</li>
<li><i>k</i> = 1/2, solid cylinder or disk around its center.</li>
</ul>
<p>For more examples, see the <a href="http://en.wikipedia.org/wiki/List_of_moments_of_inertia" title="List of moments of inertia">List of moments of inertia</a>.</p>
<p><a name="Parallel_axis_theorem" id="Parallel_axis_theorem"></a></p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Moment_of_inertia&amp;action=edit&amp;section=5" title="Edit section: Parallel axis theorem">edit</a>]</span> <span class="mw-headline">Parallel axis theorem</span></h3>
<dl>
<dd>
<div class="noprint relarticle mainarticle"><i>Main article: <a href="http://en.wikipedia.org/wiki/Parallel_axis_theorem" title="Parallel axis theorem">Parallel axis theorem</a></i></div>
</dd>
</dl>
<p>Once the moment of inertia has been calculated for rotations about the <a href="http://en.wikipedia.org/wiki/Center_of_mass" title="Center of mass">center of mass</a>
of a rigid body, one can conveniently recalculate the moment of inertia
for all parallel rotation axes as well, without having to resort to the
formal definition. If the axis of rotation is displaced by a distance <span class="texhtml"><i>R</i></span>
from the center of mass axis of rotation (e.g. spinning a disc about a
point on its periphery, rather than through its center,) the displaced
and center-moment of inertia are related as follows:</p>
<dl>
<dd><img class="tex" alt=" I_{\mathrm{displaced}} = I_{\mathrm{center}}  + M R^{2} \,\! " src="Moment_of_inertia_files/9564e64aebcc7c3843c5ec3d131e2bbc.png"></dd>
</dl>
<p>This theorem is also known as the <i>parallel axes rule</i> and is a special case of <i>Steiner's parallel-axis theorem</i>.</p>
<p><a name="Composite_bodies" id="Composite_bodies"></a></p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Moment_of_inertia&amp;action=edit&amp;section=6" title="Edit section: Composite bodies">edit</a>]</span> <span class="mw-headline">Composite bodies</span></h3>
<p>If a body can be decomposed (either physically or conceptually) into
several constituent parts, then the moment of inertia of the body about
a given axis is obtained by summing the moments of inertia of each
constituent part around the same given axis<sup id="cite_ref-0" class="reference"><a href="#cite_note-0" title="">[1]</a></sup>.</p>
<p><a name="Equations_involving_the_moment_of_inertia" id="Equations_involving_the_moment_of_inertia"></a></p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Moment_of_inertia&amp;action=edit&amp;section=7" title="Edit section: Equations involving the moment of inertia">edit</a>]</span> <span class="mw-headline">Equations involving the moment of inertia</span></h3>
<p>The rotational <a href="http://en.wikipedia.org/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a> of a <a href="http://en.wikipedia.org/wiki/Rigid_body" title="Rigid body">rigid body</a> can be expressed in terms of its moment of inertia. For a system with <span class="texhtml"><i>N</i></span> point masses <span class="texhtml"><i>m</i><sub><i>i</i></sub></span> moving with speeds <span class="texhtml"><i>v</i><sub><i>i</i></sub></span>, the rotational kinetic energy <span class="texhtml"><i>T</i></span> equals</p>
<dl>
<dd><img class="tex" alt="T = \sum_{i=1}^{N} \frac{1}{2} m_{i} v_{i}^{2}\,\! = \sum_{i=1}^{N} \frac{1}{2} m_{i} (\omega r_{i})^{2} = \frac{1}{2} \sum_{i=1}^{N} m_{i} r_{i}^{2} \omega^{2} = \frac{1}{2} I \omega^{2}" src="Moment_of_inertia_files/c49a55b22e60e33a76a6de4d9ee6f332.png"></dd>
</dl>
<p>where <span class="texhtml">ω</span> is the common angular velocity (in <a href="http://en.wikipedia.org/wiki/Radian" title="Radian">radians</a> per second). The final formula <img class="tex" alt="T=\frac{1}{2} I \omega^{2}\,\!" src="Moment_of_inertia_files/9a5727603650b9886c83f1f0ab15eaab.png"> also holds for a continuous distribution of mass with a generalisation of the above derivation from a discrete summation to an <a href="http://en.wikipedia.org/wiki/Integral" title="Integral">integration</a>.</p>
<p>In the special case where the <a href="http://en.wikipedia.org/wiki/Angular_momentum" title="Angular momentum">angular momentum</a> <a href="http://en.wikipedia.org/wiki/Vector_%28spatial%29" title="Vector (spatial)">vector</a> is parallel to the <a href="http://en.wikipedia.org/wiki/Angular_velocity" title="Angular velocity">angular velocity</a> vector, one can relate them by the equation</p>
<dl>
<dd><img class="tex" alt="L = I\omega \;" src="Moment_of_inertia_files/181735f1f99c7989d95d4bcaf76a3d5d.png"></dd>
</dl>
<p>where <i>L</i> is the angular momentum and <span class="texhtml">ω</span> is the angular velocity. However, this equation does not hold in many cases of interest, such as the torque-free <a href="http://en.wikipedia.org/wiki/Precession" title="Precession">precession</a> of a rotating object, although its more general tensor form is always correct.</p>
<p>When the moment of inertia is constant, one can also relate the <a href="http://en.wikipedia.org/wiki/Torque" title="Torque">torque</a> on an object and its angular <a href="http://en.wikipedia.org/wiki/Acceleration" title="Acceleration">acceleration</a> in a similar equation:</p>
<dl>
<dd><img class="tex" alt="\tau = I\alpha \!" src="Moment_of_inertia_files/41ebce3186374687205dc1c7d1662f84.png"></dd>
</dl>
<p>where <i><span class="texhtml">τ</span></i> is the torque and <span class="texhtml">α</span> is the angular acceleration.</p>
<p><a name="Moment_of_inertia_tensor" id="Moment_of_inertia_tensor"></a></p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Moment_of_inertia&amp;action=edit&amp;section=8" title="Edit section: Moment of inertia tensor">edit</a>]</span> <span class="mw-headline">Moment of inertia tensor</span></h2>
<p>For the same object, different axes of rotation will have different
moments of inertia about those axes. In general, the moments of inertia
are not equal unless the object is symmetric about all axes. The <b>moment of inertia <a href="http://en.wikipedia.org/wiki/Tensor" title="Tensor">tensor</a></b>
is a convenient way to summarize all moments of inertia of an object
with one quantity. It may be calculated with respect to any point in
space, although for practical purposes the center of mass is most
commonly used.</p>
<p><a name="Definition_2" id="Definition_2"></a></p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Moment_of_inertia&amp;action=edit&amp;section=9" title="Edit section: Definition">edit</a>]</span> <span class="mw-headline">Definition</span></h3>
<p>For a rigid object of <span class="texhtml"><i>N</i></span> point masses <span class="texhtml"><i>m</i><sub><i>k</i></sub></span>, the moment of inertia <a href="http://en.wikipedia.org/wiki/Tensor" title="Tensor">tensor</a> is given by</p>
<dl>
<dd><img class="tex" alt="\mathbf{I} = \begin{bmatrix}
I_{xx} &amp; I_{xy} &amp; I_{xz} \\
I_{yx} &amp; I_{yy} &amp; I_{yz} \\
I_{zx} &amp; I_{zy} &amp; I_{zz}
\end{bmatrix}" src="Moment_of_inertia_files/ac0bef65126e1debaa9b69f31c9c483d.png">.</dd>
</dl>
<p>Its components are defined as</p>
<dl>
<dd><img class="tex" alt="I_{ij} \ \stackrel{\mathrm{def}}{=}\  \sum_{k=1}^{N} m_{k} (r_k^{2}\delta_{ij} - r_{ki}r_{kj})\,\!" src="Moment_of_inertia_files/117b5f8539b140afea65eb1a0caa4ed4.png"></dd>
</dl>
<p>where</p>
<dl>
<dd><i>i</i>, <i>j</i> equal 1, 2, or 3 for x, y, and z, respectively,</dd>
<dd><i>r</i><sub><i>k</i></sub> is the distance of mass <i>k</i> from the point about which the tensor is calculated, and</dd>
<dd><i><span class="texhtml">δ<sub><i>i</i><i>j</i></sub></span></i> is the <a href="http://en.wikipedia.org/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a>.</dd>
</dl>
<p>The diagonal elements are more succinctly written as</p>
<dl>
<dd><img class="tex" alt="I_{xx} \ \stackrel{\mathrm{def}}{=}\  \sum_{k=1}^{N} m_{k} (y_{k}^{2}+z_{k}^{2}),\,\! " src="Moment_of_inertia_files/cdedcdea7c44bda01a36a3b88373cf89.png"></dd>
<dd><img class="tex" alt="I_{yy} \ \stackrel{\mathrm{def}}{=}\  \sum_{k=1}^{N} m_{k} (x_{k}^{2}+z_{k}^{2}),\,\!" src="Moment_of_inertia_files/f2e4c11bc67b4178fa771510178618ca.png"></dd>
<dd><img class="tex" alt="I_{zz} \ \stackrel{\mathrm{def}}{=}\  \sum_{k=1}^{N} m_{k} (x_{k}^{2}+y_{k}^{2}),\,\!" src="Moment_of_inertia_files/9ee2d3d5f5e85f8891e37eea0a6fa218.png"></dd>
</dl>
<p>while the off-diagonal elements, also called the <b>products of inertia</b>, are</p>
<dl>
<dd><img class="tex" alt="I_{xy} = I_{yx} \ \stackrel{\mathrm{def}}{=}\  -\sum_{k=1}^{N} m_{k} x_{k} y_{k},\,\!" src="Moment_of_inertia_files/360e798731e3d26b6e47a6b230dc60c5.png"></dd>
<dd><img class="tex" alt="I_{xz} = I_{zx} \ \stackrel{\mathrm{def}}{=}\  -\sum_{k=1}^{N} m_{k} x_{k} z_{k},\,\!" src="Moment_of_inertia_files/076200502a5bafd62ea93805de62b973.png"> and</dd>
<dd><img class="tex" alt="I_{yz} = I_{zy} \ \stackrel{\mathrm{def}}{=}\  -\sum_{k=1}^{N} m_{k} y_{k} z_{k},\,\!" src="Moment_of_inertia_files/26e1f1b07fd58400cd22de2a2bf18898.png"></dd>
</dl>
<p>Here <span class="texhtml"><i>I</i><sub><i>x</i><i>x</i></sub></span> denotes the moment of inertia around the <span class="texhtml"><i>x</i></span>-axis when the objects are rotated around the x-axis, <span class="texhtml"><i>I</i><sub><i>x</i><i>y</i></sub></span> denotes the moment of inertia around the <span class="texhtml"><i>y</i></span>-axis when the objects are rotated around the <span class="texhtml"><i>x</i></span>-axis, and so on.</p>
<p>These quantities can be generalized to an object with continuous
density in a similar fashion to the scalar moment of inertia. One then
has</p>
<dl>
<dd><img class="tex" alt="\mathbf{I}=\iiint_V  \rho(x,y,z)\left( r^2 \mathbf{E}_{3} - \mathbf{r}\otimes \mathbf{r}\right)\, dx\,dy\,dz," src="Moment_of_inertia_files/3a6687ae032f92d9df4dd4416e578956.png"></dd>
</dl>
<p>where <img class="tex" alt="\mathbf{r}\otimes \mathbf{r}" src="Moment_of_inertia_files/ccf3d5083ad66868c629ff57cdf59f41.png"> is their <a href="http://en.wikipedia.org/wiki/Outer_product" title="Outer product">outer product</a>, <b>E</b><sub>3</sub> is the 3 × 3 <a href="http://en.wikipedia.org/wiki/Identity_matrix" title="Identity matrix">identity matrix</a>, and <i>V</i> is a region of space completely containing the object.</p>
<p><a name="Derivation_of_the_tensor_components" id="Derivation_of_the_tensor_components"></a></p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Moment_of_inertia&amp;action=edit&amp;section=10" title="Edit section: Derivation of the tensor components">edit</a>]</span> <span class="mw-headline">Derivation of the tensor components</span></h3>
<p>The distance <span class="texhtml"><i>r</i></span> of a particle at <img class="tex" alt="\mathbf{x}" src="Moment_of_inertia_files/3c66d9170d4c3fb75456e1a9fc6ead37.png"> from the axis of rotation passing through the origin in the <img class="tex" alt="\mathbf{\hat{n}}" src="Moment_of_inertia_files/18049261a6aedb1158ebb8dfe6d1217b.png"> direction is <img class="tex" alt=" |\mathbf{x}-(\mathbf{x} \cdot \mathbf{\hat{n}}) \mathbf{\hat{n}}|" src="Moment_of_inertia_files/2d3ae2e6e3be4b3da74f282ea9dcd268.png">. By using the formula <span class="texhtml"><i>I</i> = <i>m</i><i>r</i><sup>2</sup></span>
(and some simple vector algebra) it can be seen that the moment of
inertia of this particle (about the axis of rotation passing through
the origin in the <img class="tex" alt="\mathbf{\hat{n}}" src="Moment_of_inertia_files/18049261a6aedb1158ebb8dfe6d1217b.png"> direction) is <img class="tex" alt=" 
I=m(|\mathbf{x}|^2 (\mathbf{\hat{n}} \cdot \mathbf{\hat{n}})-(\mathbf{x} \cdot \mathbf{\hat{n}})^2)." src="Moment_of_inertia_files/31587bfdb07dfb85116b39d4d561e100.png"> This is a <a href="http://en.wikipedia.org/wiki/Quadratic_form" title="Quadratic form">quadratic form</a> in <img class="tex" alt="\mathbf{\hat{n}}" src="Moment_of_inertia_files/18049261a6aedb1158ebb8dfe6d1217b.png"> and, after a bit more algebra, this leads to a tensor formula for the moment of inertia</p>
<dl>
<dd><img class="tex" alt="{I} = m [n_1,n_2,n_3]\begin{bmatrix}
 y^2+z^2 &amp; -xy &amp; -xz \\
-y x &amp; x^2+z^2 &amp; -yz \\
-zx &amp; -zy &amp; x^2+y^2
\end{bmatrix} \begin{bmatrix}
 n_1 \\
 n_2\\
n_3
\end{bmatrix}" src="Moment_of_inertia_files/dacd5f0fa6ccbb3553e0d656e4a58b2f.png">.</dd>
</dl>
<p>This is exactly the formula given below for the moment of inertia in
the case of a single particle. For multiple particles we need only
recall that the moment of inertia is additive in order to see that this
formula is correct.</p>
<p><a name="Reduction_to_scalar" id="Reduction_to_scalar"></a></p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Moment_of_inertia&amp;action=edit&amp;section=11" title="Edit section: Reduction to scalar">edit</a>]</span> <span class="mw-headline">Reduction to scalar</span></h3>
<p>For any axis <img class="tex" alt="\hat{\mathrm{n}}" src="Moment_of_inertia_files/7bc0acd59c12d2097b500bd52cdebb7b.png">, represented as a column vector with elements <i>n</i><sub>i</sub>, the scalar form <i>I</i> can be calculated from the tensor form <b>I</b> as</p>
<dl>
<dd><img class="tex" alt="I = \mathbf{\hat{n}^{T}} \mathbf{I}\, \mathbf{\hat{n}} = 
\sum_{j=1}^{3} \sum_{k=1}^{3} n_{j} I_{jk} n_{k} ." src="Moment_of_inertia_files/f2f62c9c20b8b0856fa3a039bb7578cb.png"></dd>
</dl>
<p>The range of both summations correspond to the three <a href="http://en.wikipedia.org/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a>.</p>
<p>The following equivalent expression avoids the use of transposed
vectors which are not supported in maths libraries because internally
vectors and their transpose are stored as the same linear array,</p>
<dl>
<dd><img class="tex" alt="I = \mathbf{{I}^{T}} \mathbf{\hat{n}} \cdot \mathbf{\hat{n}}." src="Moment_of_inertia_files/c18b4bf9e0bf25062863733b1e73d52a.png"></dd>
</dl>
<p>However it should be noted that although this equation is
mathematically equivalent to the equation above for any matrix, inertia
tensors are symmetrical. This means that it can be further simplified
to:</p>
<dl>
<dd><img class="tex" alt="I = \mathbf{{I}} \mathbf{\hat{n}} \cdot \mathbf{\hat{n}}." src="Moment_of_inertia_files/ffdd054ef67e321aaf4f726f1bff908a.png"></dd>
</dl>
<p><a name="Principal_moments_of_inertia" id="Principal_moments_of_inertia"></a></p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Moment_of_inertia&amp;action=edit&amp;section=12" title="Edit section: Principal moments of inertia">edit</a>]</span> <span class="mw-headline">Principal moments of inertia</span></h3>
<p>Since the moment of inertia tensor is real and <a href="http://en.wikipedia.org/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric</a>, it is possible to find a Cartesian coordinate system in which it is <a href="http://en.wikipedia.org/wiki/Diagonalizable_matrix" title="Diagonalizable matrix">diagonal</a>, having the form</p>
<dl>
<dd><img class="tex" alt="\mathbf{I} = \begin{bmatrix}
I_{1} &amp; 0 &amp; 0 \\
0 &amp; I_{2} &amp; 0 \\
0 &amp; 0 &amp; I_{3}
\end{bmatrix}" src="Moment_of_inertia_files/174d7736506fe6f7f6bfbbb5560c1089.png"></dd>
</dl>
<p>where the coordinate axes are called the <b><a href="http://en.wikipedia.org/wiki/Principal_axes" class="mw-redirect" title="Principal axes">principal axes</a></b> and the constants <span class="texhtml"><i>I</i><sub>1</sub></span>, <span class="texhtml"><i>I</i><sub>2</sub></span> and <span class="texhtml"><i>I</i><sub>3</sub></span> are called the <b>principal moments of inertia</b>. The unit vectors along the principal axes are usually denoted as <img class="tex" alt="(\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3})" src="Moment_of_inertia_files/7132f27664b4801d272f017d92937620.png">.</p>
<p>When all principal moments of inertia are distinct, the principal
axes are uniquely specified. If two principal moments are the same, the
rigid body is called a <b>symmetrical top</b> and there is no unique
choice for the two corresponding principal axes. If all three principal
moments are the same, the rigid body is called a <b>spherical top</b>
(although it need not be spherical) and any axis can be considered a
principal axis, meaning that the moment of inertia is the same about
any axis.</p>
<p>The principal axes are often aligned with the object's symmetry axes. If a rigid body has an axis of symmetry of order <span class="texhtml"><i>m</i></span>, i.e., is symmetrical under rotations of 360°/<i>m</i> about a given axis, the symmetry axis is a principal axis. When <span class="texhtml"><i>m</i> &gt; 2</span>,
the rigid body is a symmetrical top. If a rigid body has at least two
symmetry axes that are not parallel or perpendicular to each other, it
is a spherical top, e.g., a cube or any other <a href="http://en.wikipedia.org/wiki/Platonic_solid" title="Platonic solid">Platonic solid</a>. A practical example of this mathematical phenomenon is the routine automotive task of <a href="http://en.wikipedia.org/wiki/Tire_balance" title="Tire balance">balancing a tire</a>,
which basically means adjusting the distribution of mass of a car wheel
such that its principal axis of inertia is aligned with the axle so the
wheel does not wobble.</p>
<p><a name="Parallel_axis_theorem_2" id="Parallel_axis_theorem_2"></a></p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Moment_of_inertia&amp;action=edit&amp;section=13" title="Edit section: Parallel axis theorem">edit</a>]</span> <span class="mw-headline">Parallel axis theorem</span></h3>
<p>Once the moment of inertia tensor has been calculated for rotations about the <a href="http://en.wikipedia.org/wiki/Center_of_mass" title="Center of mass">center of mass</a> of the rigid body, there is a useful labor-saving method to compute the tensor for rotations offset from the center of mass.</p>
<p>If the axis of rotation is displaced by a vector <b>R</b> from the center of mass, the new moment of inertia tensor equals</p>
<dl>
<dd><img class="tex" alt="\mathbf{I}^{\mathrm{displaced}} = \mathbf{I}^{\mathrm{center}} + M \left[ \left(\mathbf{R} \cdot \mathbf{R}\right) \mathbf{E}_{3} - \mathbf{R} \otimes \mathbf{R} \right]" src="Moment_of_inertia_files/2c861957dd3f40ee86ebef4e2e1b4de9.png"></dd>
</dl>
<p>where <span class="texhtml"><i>M</i></span> is the total mass of the rigid body, <b>E</b><sub>3</sub> is the 3 × 3 <a href="http://en.wikipedia.org/wiki/Identity_matrix" title="Identity matrix">identity matrix</a>, and <img class="tex" alt="\otimes" src="Moment_of_inertia_files/e9dd9013ec300ceba41484dfc2c9a876.png"> is the <a href="http://en.wikipedia.org/wiki/Outer_product" title="Outer product">outer product</a>.</p>
<p><a name="Other_mechanical_quantities" id="Other_mechanical_quantities"></a></p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Moment_of_inertia&amp;action=edit&amp;section=14" title="Edit section: Other mechanical quantities">edit</a>]</span> <span class="mw-headline">Other mechanical quantities</span></h3>
<p>Using the tensor <b>I</b>, the kinetic energy can be written as a quadratic form</p>
<dl>
<dd><img class="tex" alt="T = \frac{1}{2} \boldsymbol\omega^T \mathbf{I}\, \boldsymbol\omega = 
\frac{1}{2} I_{1} \omega_{1}^{2} + \frac{1}{2} I_{2} \omega_{2}^{2} + \frac{1}{2} I_{3} \omega_{3}^{2}" src="Moment_of_inertia_files/764f402d55a4e9e16c5eeaa620c3683d.png"></dd>
</dl>
<p>and the angular momentum can be written as a product</p>
<dl>
<dd><img class="tex" alt="\mathbf{L} = \mathbf{I}\, \boldsymbol\omega = 
\omega_{1} I_{1} \mathbf{e}_{1} + \omega_{2} I_{2} \mathbf{e}_{2} + \omega_{3} I_{3} \mathbf{e}_{3}" src="Moment_of_inertia_files/1ff43a332f385a753087638ac7075117.png"></dd>
</dl>
<p>Taken together, one can express the rotational kinetic energy in terms of the angular momentum <span class="texhtml">(<i>L</i><sub>1</sub>,<i>L</i><sub>2</sub>,<i>L</i><sub>3</sub>)</span> in the principal axis frame as</p>
<dl>
<dd><img class="tex" alt="T = 
\frac{L_{1}^{2}}{2I_{1}} + \frac{L_{2}^{2}}{2I_{2}} + \frac{L_{3}^{2}}{2I_{3}}.\,\!" src="Moment_of_inertia_files/318d8ad54e1080d2f96384cd50d0ec30.png"></dd>
</dl>
<p>The rotational kinetic energy and the angular momentum are constants
of the motion (conserved quantities) in the absence of an overall <a href="http://en.wikipedia.org/wiki/Torque" title="Torque">torque</a>. The angular velocity ω is <i>not constant</i>; even without a torque, the endpoint of this vector may move in a plane (see <a href="http://en.wikipedia.org/wiki/Poinsot%27s_construction" class="mw-redirect" title="Poinsot's construction">Poinsot's construction</a>).</p>
<p><i>See the article on the <a href="http://en.wikipedia.org/wiki/Rigid_rotor#Classical_kinetic_energy" title="Rigid rotor">rigid rotor</a> for more ways of expressing the kinetic energy of a <a href="http://en.wikipedia.org/wiki/Rigid_body" title="Rigid body">rigid body</a>.</i></p>
<p><a name="See_also" id="See_also"></a></p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Moment_of_inertia&amp;action=edit&amp;section=15" title="Edit section: See also">edit</a>]</span> <span class="mw-headline">See also</span></h2>
<ul>
<li><a href="http://en.wikipedia.org/wiki/List_of_moments_of_inertia" title="List of moments of inertia">List of moments of inertia</a></li>
<li><a href="http://en.wikipedia.org/wiki/List_of_moment_of_inertia_tensors" title="List of moment of inertia tensors">List of moment of inertia tensors</a></li>
<li><a href="http://en.wikipedia.org/wiki/Rotational_energy" title="Rotational energy">Rotational energy</a></li>
<li><a href="http://en.wikipedia.org/wiki/Parallel_axis_theorem" title="Parallel axis theorem">Parallel axis theorem</a></li>
<li><a href="http://en.wikipedia.org/wiki/Perpendicular_axis_theorem" title="Perpendicular axis theorem">Perpendicular axis theorem</a></li>
<li><a href="http://en.wikipedia.org/wiki/Stretch_rule" title="Stretch rule">Stretch rule</a></li>
<li><a href="http://en.wikipedia.org/wiki/Poinsot%27s_ellipsoid" title="Poinsot's ellipsoid">Poinsot's ellipsoid</a></li>
</ul>
<p><a name="References" id="References"></a></p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Moment_of_inertia&amp;action=edit&amp;section=16" title="Edit section: References">edit</a>]</span> <span class="mw-headline">References</span></h2>
<ul>
<li>Goldstein H. (1980) <i>Classical Mechanics</i>, 2nd. ed., Addison-Wesley. <a href="http://en.wikipedia.org/wiki/Special:BookSources/0201029189" class="internal">ISBN 0-201-02918-9</a></li>
</ul>
<ul>
<li>Landau LD and Lifshitz EM. (1976) <i>Mechanics</i>, 3rd. ed., Pergamon Press. <a href="http://en.wikipedia.org/wiki/Special:BookSources/0080210228" class="internal">ISBN 0-08-021022-8</a> (hardcover) and <a href="http://en.wikipedia.org/wiki/Special:BookSources/0080291414" class="internal">ISBN 0-08-029141-4</a> (softcover).</li>
</ul>
<ul>
<li>Marion JB and Thornton ST. (1995) <i>Classical Dynamics of Systems and Particles</i>, 4th. ed., Thomson. <a href="http://en.wikipedia.org/wiki/Special:BookSources/0030973023" class="internal">ISBN 0-03-097302-3</a></li>
</ul>
<ul>
<li>Symon KR. (1971) <i>Mechanics</i>, 3rd. ed., Addison-Wesley. <a href="http://en.wikipedia.org/wiki/Special:BookSources/0201073927" class="internal">ISBN 0-201-07392-7</a></li>
</ul>
<ul>
<li>Tenenbaum, RA. (2004) <i>Fundamentals of Applied Dynamics</i>, Springer. <a href="http://en.wikipedia.org/wiki/Special:BookSources/038700887X" class="internal">ISBN 0-387-00887-X</a></li>
</ul>
<p><a name="External_links" id="External_links"></a></p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Moment_of_inertia&amp;action=edit&amp;section=17" title="Edit section: External links">edit</a>]</span> <span class="mw-headline">External links</span></h2>
<ul>
<li><a href="http://www.lightandmatter.com/html_books/0sn/ch04/ch04.html" class="external text" title="http://www.lightandmatter.com/html_books/0sn/ch04/ch04.html" rel="nofollow">Angular momentum and rigid-body rotation in two and three dimensions</a></li>
<li><a href="http://www.physics.uoguelph.ca/tutorials/torque/Q.torque.inertia.html" class="external text" title="http://www.physics.uoguelph.ca/tutorials/torque/Q.torque.inertia.html" rel="nofollow">A table of moments of inertia</a></li>
<li><a href="http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html" class="external text" title="http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html" rel="nofollow">Lecture notes on rigid-body rotation and moments of inertia</a></li>
<li><a href="http://kwon3d.com/theory/moi/iten.html" class="external text" title="http://kwon3d.com/theory/moi/iten.html" rel="nofollow">The moment of inertia tensor</a></li>
<li><a href="http://www.phy.hk/wiki/englishhtm/Balance.htm" class="external text" title="http://www.phy.hk/wiki/englishhtm/Balance.htm" rel="nofollow">An introductory lesson on moment of inertia: keeping a vertical pole not falling down (Java simulation)</a></li>
</ul>


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